# Thread: Inequality Application

1. ## Inequality Application

A charter airline finds that on its Saturday flights from Philadelphia to London, all 120 seats will be sold if the ticket price is $200. However, for each$3 increase in ticket price, the number of seats sold decreases by one.

a) Find a formula for the number of seats sold if the ticket price is P dollars.
The answer for a is $\displaystyle -\frac{1}{3}P + \frac{560}{3}$. However, I do not see how they arrived at that answer. Any help will be greatly appreciated.

2. Originally Posted by mathgeek777
The answer for a is $\displaystyle -\frac{1}{3}P + \frac{560}{3}$. However, I do not see how they arrived at that answer. Any help will be greatly appreciated.
do you realize that we have a linear relationship between the number of seats filled and the ticket price? since each changes at a constant rate with respect to each other.

lets think of this graphically so that you get the idea. we want a function of price, so put that on the x-axis. the number of seats on the y-axis. now, plot a few points:

$\displaystyle \begin{array}{c|c} S & P \\ \hline 120 & 200 \\ 119 & 203 \\ 118 & 206 \\ . & . \\ . & . \\ . & . \end{array}$

S is the number of seats, P is the price. S goes down by 1 when P goes up by 3

all we need to do is find the straight line that passes through these points. can you do that?

3. Ok. I see where the $\displaystyle -\frac{1}{3}$ came from, but i still don't know how you get $\displaystyle \frac{560}{3)$

4. Originally Posted by mathgeek777
Ok. I see where the $\displaystyle -\frac{1}{3}$ came from, but i still don't know how you get $\displaystyle \frac{560}{3}$
the equation of a line in the slope-intercept form is $\displaystyle y = mx + b$, where $\displaystyle m$ is the slope and $\displaystyle b$, is the y-intercept.

given any two points on the line, $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2,y_2)$, we can find the equation of the line as follows:

$\displaystyle m = \frac {y_2 - y_1}{x_2 - x_1}$

By the point-slope form, plug in the values of $\displaystyle m$, $\displaystyle x_1$ and $\displaystyle y_1$ into:

$\displaystyle y - y_1 = m(x - x_1)$

and solve for $\displaystyle y$ to get it into the slope-intercept form.

alternatively, we could plug the points into $\displaystyle y = mx + b$ and solve for $\displaystyle b$, since it would be the only unknown at this point

here of course, your y is S and your x is P

5. Hello, mathgeek777!

A charter airline finds that on its Saturday flights from Philadelphia to London,
all 120 seats will be sold if the ticket price is $200. However, for each$3 increase in ticket price, the number of seats sold decreases by one.

a) Find a formula for the number of seats sold if the ticket price is $\displaystyle P$ dollars.

The answer is: $\displaystyle -\frac{1}{3}P + \frac{560}{3}$

Let $\displaystyle x$ = number of $3-increases in the price of a ticket. Then the price is: .$\displaystyle P \:=\:200 + 3
x\quad\Rightarrow\quad x \:=\:\frac{P - 200}{3}\;\;{\color{blue}[1]}$. . and the number of seats sold is: .$\displaystyle S \;=\;120 - x\;\;{\color{blue}[2]}$Substitute [1] into [2]: .$\displaystyle S \;=\;120 - \frac{P-200}{3} \;=\;\frac{360}{3} - \frac{P-200}{3} \;=\;\frac{-P + 560}{3}$Therefore: .$\displaystyle S \;=\;-\frac{1}{3}P + \frac{560}{3}$6. ## Re: Inequality Application Originally Posted by Jhevon the equation of a line in the slope-intercept form is$\displaystyle y = mx + b$, where$\displaystyle m$is the slope and$\displaystyle b$, is the y-intercept. given any two points on the line,$\displaystyle (x_1, y_1)$and$\displaystyle (x_2,y_2)$, we can find the equation of the line as follows:$\displaystyle m = \frac {y_2 - y_1}{x_2 - x_1}$..... That thread is very old, but I dont understand something and I have a question why the slope is$\displaystyle m = \frac {-1}{3}$? I used the formula of the slope and I used also the values of the table then:$\displaystyle m = \frac {203-200}{119-120}$the answer is$\displaystyle m=-3$and not$\displaystyle \frac {-1}{3}$Sincerely thanks, 7. ## Re: Inequality Application Originally Posted by birthdayalex That thread is very old, but I dont understand something and I have a question why the slope is$\displaystyle m = \frac {-1}{3}$? I used the formula of the slope and I used also the values of the table then:$\displaystyle m = \frac {203-200}{119-120}$the answer is$\displaystyle m=-3$and not$\displaystyle \frac {-1}{3}\$
For some reason, JHevon's chart is mapping seats to prices when the problem is mapping prices to seats. He switched the x- and y- axes (giving you slopes that are inverses of the ones in the problem). The slope should be:

$$m = \dfrac{119-120}{203-200} = -\dfrac{1}{3}$$